We consider the exact iterative algorithm for solution of the direct and inverse scattering problems in 1D inhomogeneous half-space using piecewise-constant approximation of the constitutive parameter profiles (e.g., refractive index). A novel modification of the Gelfand-Levitan equation is derived for solution of the inverse scattering problem. Numerical examples are given and the effects related to a finite frequency band of the sounding pulse are discussed.